Constantin C. Brîncuş
March 01 until August 31, 2025
Affiliation: Institute of Philosophy and Psychology, Romanian Academy // Faculty of Philosophy, University of Bucharest
Research for a study about:
From Hilbert to Carnap and beyond: Categoricity by Inferential Conservativity
The main objective of this project is to develop a new solution to the problem of categoricity, i.e. of uniquely determining by proof-theoretic means the standard model of a formal theory. Unlike other recent proposals, my solution draws on the historical and philosophical background of Hilbert’s formalism and the Vienna Circle logical empiricism.
Due to Gödel’s results, the categoricity of a theory and the semantic completeness of its underlying logic are actually mutually exclusive. In second-order logic (SOL) we obtain the categoricity of Peano Arithmetic (PA), but SOL with standard models (in which PA is categorical) is semantically incomplete. In first-order logic we have semantic completeness, but all first-order theories that allow infinite models are non-categorical. Moreover, the syntactic completeness of PA is lost in both logics. One way to obtain syntactic completeness is by using Hilbert’s new rule of inference, i.e. the ω-rule, as Carnap did. The omega logic is semantically complete and PA with omega logic (PAω) is syntactically complete. What is still lost is the categoricity of PA.
One way to obtain the categoricity of PAω is by adding supplementary constraints. The argument for the categoricity of PAω that I aim to develop uses a proof-theoretic notion of conservativity that goes back to Carnap, Belnap, Dummett and Brandom, i.e. inferential conservativity. In this sense, a theory or a system of rules is inferentially conservative if and only if no extension of the original language allows new (material) inferences in the old system. This notion of conservativity is inferential since it assumes that the meaning of an expression is determined by the inferential use of that expression. The material inferences are those that determine the meanings of the non-logical expressions. The argument that I want to develop aims to establish that if PAω is inferentially conservative, then it is categorical.
Lecture
Categorical countable structures in ω-logic and Lω1ω
Logik Café Lecture
Date: May 19, 2025
Time: 4.30 pm - 6 pm
Meetings are usually held on Mondays from 16:30 to 18:00 in Room 3B (D0315), 3. floor Department of Philosophy, NIG/ Neues Institutsgebäude, Universitätsstraße 7, 1010 Vienna
Abstract:
I argue in this presentation that if a sentence φ from Lω1ω is categorical, then the first-order theory Tφ that corresponds to φ is categorical if and only if its models satisfy the ω-rule, inferentially understood. This result will be established by first showing that if all the models of Peano Arithmetic in the ω-logic, i.e. PAω, have the property expressed by the ω-rule, then PAω is categorical. The presentation is based on joint work with John T. Baldwin.
Vortrag im Rahmen des Workshops “From Permanence to Open-endedness”
Report
The main aim of my fellowship was to develop a new solution to the problem of categoricity for first-order theories and their underlying logics, starting from the historical and philosophical background of Hilbert’s formalism and Carnap’s work on the foundations of logic and mathematics. The research stay was extremely productive, leading to the development of a new model-theoretic inferentialist account of the categoricity of first-order theories, which I worked on together with John T. Baldwin, professor emeritus of mathematics, statistics, and computer science at the University of Illinois at Chicago. I presented some of the results at the following conferences: Logik Café, Institut Wiener Kreis // International Workshop: “From Permanence to Open-endedness,” Department of Philosophy, University of Vienna (May 19–20, 2025), and at the CIVIS Summer School: Let’s Prove It! Reasoning in Different Contexts, National and Kapodistrian University of Athens (July 7–11, 2025).
I also attended presentations given at conferences and workshops at the Institut Wiener Kreis (especially at Logik Café), which I found—along with the subsequent discussions—very helpful for my project and for understanding the current state of research in the philosophy of logic and mathematics. I had highly productive interactions with colleagues involved in the Institut Wiener Kreis or attending specific conferences, especially William Agay-Beaujon, Georg Brun, Mirko Engler, Christoph Limbeck-Lilienau, Georg Schiemer, Sebastian G. W. Speitel, Marta Sznajder, Jennifer Whyte, and Benjamin Zayton.